η- Ricci solitons on lorentzian para- sasakian manifolds defined with W2−curvaturetensor

In the present paper η- Ricci solitons on Lorentzian Para- Sasakian manifolds satisfying (ξ,.)s.W2 = 0 and (ξ,.)W2.S = 0 are treated. The results obtained on η- Ricci solitons on para- Kenmotsu manifolds have motivated us to investigate η- Ricci solitons on Lorentzian Para-Sasakian Manifolds satisfying the same conditions and quasi-similar results have been obtained. In fact, we have proved that Lorentzian Para- Sasakian manifolds satisfying (ξ,.) s.W2 = 0 and having η− Ricci soliton structure are Einstein or quasi-Einstein manifolds according to the value µ and λ. The same results have been proved on Manifolds satisfying (ξ,.)W2.S = 0.

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Introduction

1.1 Background In last years the Ricci solitons have interested most of geometers as topic of study on different manifolds. Contact and para-Contact have been among the most considered in the study of those solutions of Ricci flows. The interest in study has considerably motivated by the recent Perelman proof of Poincar´e Conjecture using Ricci flows. The Ricci Flows have been introduced by R.S as generalization of Einstein metrics in 1982 [1].

Ricci flow is an evolution equation of heat equation type for the metric on Reimannian manifold and is defined as (1.1) 1.2 Ricci soliton A Ricci soliton (g, v, λ) on Reimannian manifold (M, g) is given by LV g + 2S + 2λg = 0 (1.2) Where S is the Ricci tensor, LV is the Lie derivative operator on M in direction V and λ is a scalar. Ricci solitons have been discussed on different manifolds such us: K¨ahler manifolds [2], On Para - Sasakian manifolds [3], on Kenmotsu [4] and f -Kenmotsu Manifolds [5] and more manifolds have been discussed by various authors. A general notion of η− Ricci soliton has been introduced by J.T. Chao and M.

Kimura in [6] and has been studied on Hopf hypersurfaces in complex spaces by C. C˘alin in [7]. In this paper we consider η− Ricci solitons on Lorentzian para- Saasakian manifolds satisfying (ξ,.) s.W8 = 0 and (ξ,.)W8.S = 0. Note that Blaga [8] has obtained some results on Ricci solitons satisfying (ξ,.) S.R = 0 and (ξ,.) R.S = 0 on the same manifolds. 2.

Lorentzian para-kenmotsu manifolds If M is m- dimensional differentiable manifold, ϕ a (1, 1)−type tensor field, η 1−form and g a Lorentzian metric on M , then Deftnition 2.1. [8] (ϕ, ξ, η, g) is said to be Lorentzian structure on M if: ϕξ = 0, η(ϕ) = 0, (2.1) Maths 2019; 4(5): 49-54 LF Uwimbabazi Ruganzu School of Mathematics, University of Nairobi, P.O. Box 30197, Nairobi, Kenya SK Moindi School of Mathematics, University of Nairobi, P.O. Box 30197, Nairobi, Kenya GP Pokhariyal School of Mathematics, University of Nairobi, P.O. Box 30197, Nairobi, Kenya Correspondence LF Uwimbabazi Ruganzu School of Mathematics, University of Nairobi, P.O.

Box 30197, Nairobi, Kenya http://www.mathsjournal.com η(ξ) = −1, ϕ2 = I + η ⊗ ξ, (2.2) g(ϕ., ϕ.) = g + η ⊗ η, (2.3) (∇X ϕ)Y = g(X, Y)ξ + 2η(X)η(Y)ξ + η(Y)X ∀X, Y ∈ X(M) (2.4) where X(M) is algebra of vector fields on M and ∇ is the Levi- Civita connection associated to g. and η ⊗ η(X, Y) stands for η(X)η(Y) Using (2.3) and (2.1), we get g(ξ, ξ) = −1, (2.5) η(X) = g(X, ξ) (2.6) and g(ϕX, Y) = g(X, ϕY) X, Y ∈ X(M) (2.7) The immediate properties of Lorentzian structure are given in the following proposition proved in [8] Proposition 2.2. On Lorentzian Para- Sasakian manifolds, the followings hold: ∇X ξ = ϕX, (2.8) η(∇X ξ) = 0, ∇ξξ = 0, (2.9) R(X, Y)ξ = −η(X)Y + η(Y)X (2.10) η(R(X, Y)Z) = η(X)g(Y, Z) − η(y)g(X, Z), η(R(X, Y)ξ) = 0 (2.11) (∇X η)Y = (∇Y η)X = g(ϕX, Y), ∇ξη = 0, (2.12) Lξϕ = 0, Lξη = 0, Lξg = 2g(ϕ.,.) (2.13) Where R is Reimannian curvature tensor field and ∇ is Levi- Civita connection associated to g. From this proposition we get that g(X, ϕY) is symmetric and (∇g(X, ϕ)) (Y, Z) = η(Y)(X, Z) + η(Z)g(X, Y) + 2η(X)η(Y)η(Z) (2.14) and g(ϕX, ϕ2Y) = g(X, ϕY) (2.15) Let (M, ϕ, ξ, η, g) be a Lorentzian manifold. The data satisfying the equation Lξg + 2s + 2λg + 2µη ⊗ η (2.16) Where Lξ is the Lie derivative operator along the vector field ξ, S is the Ricci tensor field of the metric g, and λ and µ are scalars is said to be η−Ricci soliton on M.

Writing Lξ in terms of the Levi- Civita connection (2.16) becomes: 2S (X, Y) = −g(∇X ξ, Y) − g(X, ∇Y ξ) − 2λg(X, Y) − µη(X)η(Y) ∀X, Y ∈ X(M). (2.17) Using (2.8) in (2.16), we get S(X, Y) = −g(ϕX, Y) − λg(X, Y) − µη(X)η(Y). (2.18) In [9] Matsumoto, Koji and Mihai proved that on Lorentzian para- Sasakian manifold (M, ϕ, ξ, η, g) the Ricci tensor satisfies S(X, ξ) = (dim(M) − 1))η(X) (2.19) and S(ϕX, ϕY) = S(X, Y) + (dim(M) − 1)η(X)η(Y) (2.20) Putting Y = ξ in (2.18), we get http://www.mathsjournal.com µ − λ = 2n (2.21) for a (2n + 1)−dimensional Lorentzian Para- Sasakian manifold M. In [8], the author studied the Lorentzian Para - Sasakian Manifolds having cyclic Ricci tensor and those with cyclic η− recurrent Ricci tensor and proved that there is no Ricci soliton with potential vector field ξ. In the same paper the author discussed the Lorentzian Para- Sasakian manifolds satisfying (ξ,.) R.S = 0, and (ξ,.) S.R = 0 and proved that in such cases M is Einstein. In the present paper Lorentzian Para- Sasakian satisfying (ξ,.) W2.S and those satisfying (ξ) S.W2 are discussed.

Note that W2 curvature tensor has been introduced by by Pokhariyal and Mishra [10] and it is defined as 1 W2(X, Y) Z = R (X, Y) Z + 2n (g(X, Z)QY − S(Y, Z)QX) (2.22) for a (2n + 1)− Lorentzian Para- Sasakian Manifolds, where QX is Ricci tensor operator defined as g(QX, Y) = S(X, Y). (2.23) Using 2.18, we get QX = −ϕX − λX − µη(X)ξ. (2.24) The following definition will be used later in derivation of our results. Deftnition 2.3. A Ricci soliton is said to be quasi- Einstein if its Ricci curvature tensor field S is linear combination (with scalars λ and µ respectively, with µ different of zero) of g and the tensor product of a non- zero 1- form η satisfying η(x) = g(X, ξ), for ξ a unit vector field and respectively, Einstein if S is collinear with g. 3. η−Ricci Solitons on Lorentzian Para- Sasakian Manifolds satisfying (ξ,.) W2.S = 0. The condition to be satisfied by S is S(W2(ξ, X)Y, Z) + S(Y, W2(ξ, X)Z) = 0 (3.1) S(W2(ξ, X)Y, Z) = η(Y){g(ϕX, Z) + λg(X, Z) − µη(X)η(Z) 1 + [g(X, Z) + η(X)η(Z) + 2λg(ϕX, Z) + λ2g(X, Z) + 2λµη(X)η(Z) µ2η(X)η(Z)]. (3.2) 2n That is 1 S (W2(ξ, X)Y, Z) = 2n [(λ + µ)g(ϕX, Z)η(Y) + (λµ + 1)(g(X, Z)η(Y) + η(x)η(Y)η(Z))] (3.3) and 1 S (Y, W (ξ, X)Z) = [(λ + µ)g(ϕX, Y)η(Z) + (λµ + 1)(g(X, Y)η(Z) + η(x)η(Y)η(Z))] (3.4) 2n Putting Z = ξ, the condition (3.1) becomes (λ + µ)g(ϕX, Y) + (λµ + 1)g(ϕX, ϕY) = 0 (3.5) Putting Y = ϕY in (3.5), we get: (λ + µ)g(ϕX, ϕY) + (λµ + 1)g(ϕX, Y) = 0. (3.6) Adding (3.5) and (3.6), we get: (µ + λ + λµ + 1)(g(ϕX, ϕY) + g(ϕX, Y) = 0 (3.7) That is (λ + 1)(µ + 1) = 0 (3.8) Thus by, the following theorem is stated.

Theorem 3.1. If (ϕ, ξ, η, g) is a Lorentzian para- Sasakian structure on the (2n + 1)−dimensional manifold M satisfying (ξ,.)W2.S = 0 , (g, ξ, λ,µ) is η−Ricci soliton on M , then µ = −1 and λ = −2n − 1 or λ = −1 and µ = 2n − 1. For µ = −1 we get from 3.5 that (λµ + 1)g(ϕX, Y) = g(ϕX, ϕY) (3.9) http://www.mathsjournal.com and using 3.5 in 2.18, we get S(X, Y) = −2ng(X, X) (3.10) Thus, we have the following theorem. Theorem 3.2.

If (ϕ, ξ, η, g) is a Lorentzian para- Sasakian structure on the (2n + 1)−dimensional manifold M satisfying (ξ,.) W2.S = 0 , (g, ξ, λ,µ) is η−Ricci soliton on M , then for µ = −1 M is Einstein For λ = −1 we get from 3.5 that g(ϕX, Y) = g(ϕX, ϕY) (3.11) and from 3.5 and 2.18, we get S(X, Y) = −2nη(X)η(Y) (3.12) Thus, we have the following theorem. Theorem 3.3. If (ϕ, ξ, η, g) is a Lorentzian para- Sasakian structure on the (2n + 1)−dimensional manifold M satisfying (ξ,.) W2.S = 0, (g, ξ, λ,µ) is η−Ricci soliton on M , then for λ = −1 M is Quasi- Einstein. The following corollary is deduced from the above two theorems.

Corollary 3.4. On Lorentzian Para- Sasakian manifold (M, ϕ, ξ, η, g) satisfying (ξ,.) W2.S = 0, there is no Ricci soliton with potential vector ξ. 4 η−Ricci Solitons on Lorentzian Para-Sasakian Manifolds satisfying (ξ,.) S.W2 = 0. The condition to be satisfied by S is: S(X, W2(Y, Z)V)ξ − S(ξ, W2(Y, Z)V)X +S(X, Y)W2(ξ, Z)V − S(ξ, Y)W2(X, Z)V +S(X, Z)W2(Y, Z)V − S(ξ, Z)W2(Y, X)V +S(X, V)W2(Y, Z)ξ − S(ξ, V)W2(Y, Z)X = 0. (4.1) Taking inner product with ξ the relation (4.1) becomes: −S(X, W2(Y, Z)V) − S(ξ, W2(Y, Z)V)η(X)+ S(X, Y)η(W2(ξ, Z)V) − S(ξ, Y)η(W2(X, Z)V) +S(X, Z)η(W2(Y, ξ)V) − S(ξ, Z)η(W2(Y, X)V) +S(X, V)η(W2(Y, Z)V) − S(ξ, V)η(W2(Y, Z)X) = 0 (4.2) After using (2.18) and expanding the terms in (4.2) and putting V = Z = ξ the condition to be satisfied by S becomes: (λ + µ)g(ϕX, Y) + (λµ + 1) = 0 (4.3) Putting Y = ϕY , we get (λ + µ)g(ϕX, ϕY) + (λµ + 1)g(ϕX, Y) = 0 (4.4) After adding (4.3) and (4.4), we get (λ + λµ + µ + 1)[g(ϕX, ϕY) + g(ϕX, Y)] = 0 (4.5) Thus, the following theorem is stated. Theorem 4.1.

If (ϕ, ξ, η, g) is a Lorentzian para- Sasakian structure on the (2n + 1)−dimensional manifold M satisfying (ξ,.)S.W2 = 0 , (g, ξ, λ,µ) is η−Ricci soliton on M , then µ = −1 and λ = −2n − 1 or λ = −1 and µ = 2n − 1. For µ = −1 we get from 4.3 that (λµ + 1)g(ϕX, Y) = g(ϕX, ϕY) (4.6) and using 4.3 in 2.18, we get S(X, Y) = −2ng(X, X) (4.7) Thus, we have the following theorem. Theorem 4.2. If (ϕ, ξ, η, g) is a Lorentzian para- Sasakian structure on the (2n + 1)−dimensional manifold M satisfying (ξ,.)S.W2 = 0 , (g, ξ, λ,µ) is η−Ricci soliton on M , then for µ = −1 M is Einstein For λ = −1 we get from 4.3 that g(ϕX, Y) = g(ϕX, ϕY) (4.8) http://www.mathsjournal.com and from 4.3 and 2.18, we get S(X, Y) = −2nη(X)η(Y) (4.9) Thus, we have the following theorem.

Theorem 4.3. If (ϕ, ξ, η, g) is a Lorentzian para- Sasakian structure on the (2n + 1)−dimensional manifold M satisfying (ξ,.)S.W2 = 0, (g, ξ, λ,µ) is η−Ricci soliton on M , then for λ = −1 M is Quasi- Einstein From the above two theorems the following corollary is deduced. Corollary 4.4. On Lorentzian Para- Sasakian manifold (M, ϕ, ξ, η, g) satisfying (ξ,.) S.W8 = 0, there is no Ricci soliton with potential vector ξ.

Conclusion

In this paper, we have proved that Lorentzian Para- Sasakian manifolds satisfying (ξ,.) s.W2 = 0 and having η− Ricci soliton structure are Einstein or quasi-Einstein manifolds according to the value µ and λ. The same results have been proved on Manifolds satisfying (ξ,.) W2.S = 0. in the future we are planning to consider the same conditions on Sasakian Manifolds and on Contact manifolds. We hope to get the interesting results.

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